In the summer 2014 term we had a talk given by Sky Brewer, a Ph.D. student here at York working under Chris Hughes and Sanju Velani on the distribution of the zeros of the Riemann zeta function, who talked to us about geometry in the complex numbers. The video is now available online for your viewing pleasure.
Applying a rotation to a circle gives you the same circle, just as applying a shear mapping to a parabola keeps the same parabola or a squeeze to a hyperbola. Using Möbius transformations and "hyper"-complex numbers we can explore these conics and maps in a rigorous algebraic way. One aspect classically known is hyperbolic geometry, famously modelled by tessellations of the circle in Escher's Circle Limits. A question to explore is: "What do these pictures look like in the lens of a hyperboloid, or a paraboloid instead of the classical sphere?"
Many thanks go to YSTV for organising these videos. You can also see it on their website here.